Full Causal Bulk Viscous Cosmological Models with Variable G and Lambda

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Full Causal Bulk Viscous Cosmological Models with Variable G and Λ

J. A. Belinchon,1, ∗ T. Harko,2, † and M. K. Mak3, ‡

1Grupo Inter-Universitario de Analisis Dimensional.Dept. Fısica ETS Arquitectura UPM Av. Juan de Herrera 4 Madrid 28040 Espana

2Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, P. R. China.3Department of Physics, The Hong Kong University of Science and Technology,Clear Water Bay, Hong Kong, P. R. China.

(Dated: February 7, 2008)

We study the evolution of a flat Friedmann-Robertson Walker Universe, filled with a bulk vis-cous cosmological fluid, in the presence of variable gravitational and cosmological constants. Thedimensional analysis of the model suggest a proportionality between the bulk viscous pressure ofthe dissipative fluid and the energy density. With the use of this assumption and with the choice ofthe standard equations of state for the bulk viscosity coefficient, temperature and relaxation time,the general solution of the field equations can be obtained, with all physical parameters having apower-law time dependence. The symmetry analysis of this model, performed by using Lie grouptechniques, confirms the unicity of the solution for this functional form of the bulk viscous pressure.

PACS numbers: 98.80.Hw, 98.80.Bp, 04.20.Jb, Variable Constants, Bulk Viscous models, Cosmology

I. INTRODUCTION

In many cosmological and astrophysical situations an idealised fluid model of the matter is inappropriate. Suchpossible situations are the relativistic transport of photons, mixtures of cosmic elementary particles, the evolution ofcosmic strings due to their interaction with each other and with the surrounding matter, the classical description ofthe (quantum) particle production phase, interaction between matter and radiation, quark and gluon plasma viscosity,different components of dark matter etc. [1]. From a physical point of view the inclusion of dissipative terms in theenergy momentum tensor of the cosmological fluid seems to be the best-motivated generalization of the matter termof the gravitational field equations.

The theories of dissipation in Eckart-Landau formulation [2], [3], who made the first attempt at creating a relativistictheory of viscosity, are now known to be pathological in several respects. Regardless of the choice of equation of state,all equilibrium states in these theories are unstable. In addition, as shown by Israel [4], signals may be propagatedthrough the fluid at velocities exceeding the speed of light. These problems arise due to the first order natureof the theory since it considers only first-order deviations from the equilibrium, leading to parabolic differentialequations, hence to infinite speeds of propagation for heat flow and viscosity, in contradiction with the principle ofcausality. While such paradoxes appear particularly glaring in relativistic theory, infinite propagation speeds alreadyconstitutes a difficulty at the classical level, since one does not expect thermal disturbances to be carried faster thansome (suitably defined) mean molecular speed. Conventional theory is thus applicable only to phenomena which are“quasi-stationary” i.e. slowly varying on space and time scales characterized by mean free path and mean collisiontime [4]. This is inadequate for many phenomena in high-energy astrophysics and relativistic cosmology involvingsteep gradients or rapid variations. These deficiencies can be traced to the fact that the conventional theories (bothclassical and relativistic) make overly restrictive hypothesis concerning the relation between the fluxes and densitiesof entropy, energy and particle number.

A relativistic second-order theory was found by Israel [4] and developed by Israel and Stewart [5] into what is called‘transient’ or ‘extended’ irreversible thermodynamics. In this model deviations from equilibrium (bulk stress, heatflow and shear stress) are treated as independent dynamical variables leading to a total of 14 dynamical fluid variablesto be determined. The solutions of the full causal theory are well behaved for all times. Hence the advantages of thecausal theories are the followings [6]: 1) for stable fluid configurations the dissipative signals propagate causally 2)unlike Eckart-type’s theories, there is no generic short-wavelength secular instability in causal theories and 3) evenfor rotating fluids, the perturbations have a well-posed initial value problem. Therefore, the best currently availabletheory for analyzing dissipative processes in the Universe is the full Israel-Stewart causal thermodynamics.

∗Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]

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Due to the complicated nonlinear character of the evolution equations, very few exact cosmological solutions of thegravitational field equations are known in the framework of the full causal theory. For a homogeneous Universe filledwith a full causal viscous fluid source obeying the relation ξ ∼ ρ

12 , exact general solutions of the field equations have

been obtained in [7], [8], [11]-[15]. In this case the evolution of the bulk viscous cosmological model can be reducedto a Painleve-Ince type differential equation. It has also been proposed that causal bulk viscous thermodynamics canmodel on a phenomenological level matter creation in the early Universe [8], [16].

Recent observations of type Ia supernovae with redshift up to about z . 1 provided evidence that we may live ina low mass-density Universe, with the contribution of the non-relativistic matter (baryonic plus dark) to the totalenergy density of the Universe of the order of Ωm ∼ 0.3 [17]-[19]. The value of Ωm is significantly less than unity [20]and consequently either the Universe is open or there is some additional energy density ρ sufficient to reach the valueΩtotal = 1, predicted by inflationary theory. Observations also show that the deceleration parameter of the Universeq is in the range −1 ≤ q < 0, and the present-day Universe undergoes an accelerated expansionary evolution.

Several physical models have been proposed to give a consistent physical interpretation to these observational facts.One candidate, and maybe the most convincing one for the missing energy is vacuum energy density or cosmologicalconstant Λ [21].

Since the pioneering work of Dirac [22], who proposed, motivated by the occurrence of large numbers in Universe, atheory with a time variable gravitational coupling constant G, cosmological models with variable G and nonvanishingand variable cosmological term have been intensively investigated in the physical literature [23]-[42]. In the isotropiccosmological model of Chen and Wu [30] it is supposed, in the spirit of quantum cosmology, that the effectivecosmological constant Λ varies as a−2 (with a the scale function). In the cosmological model of Lima and Maia [41]the cosmological constant Λ = Λ (H) = 3βH2 + 3 (1 − β)H3/HI is a complicated function of the Hubble parameterH , a constant β and an arbitrary time scale H−1

I , leading to a cosmic history beginning from an instability of the deSitter space-time. The cosmological implications of a time dependence of the cosmological of the form Λ ∼ t−2 havebeen considered by Berman [28]. Waga [42] investigated flat cosmological models with the cosmological term varyingas Λ = α/a2 + βH2 + γ, with α, β and γ constants. In this model exact expressions for observable quantities canbe obtained. Nucleosynthesis in decaying-vacuum cosmological models based on the Chen-Wu ansatz [30] has beeninvestigated by Abdel-Rahman [24]. The consistency with the observed helium abundance and baryon asymmetryallows a maximum vacuum energy close to the radiation energy today. Anisotropic Bianchi type I cosmologicalmodels with variable G and Λ have been analyzed by Beesham [27] and it was shown that in this case there are noclassical inflationary solutions of pure exponential form. Cosmological models with the gravitational and cosmologicalconstants generalized as coupling scalars and with G ∼ an have been discussed by Sistero [40]. Generalized fieldequations with time dependent G and Λ have been proposed in [35] and [36] in an attempt to reconcile the largenumber hypothesis with Einstein’s theory of gravitation. Limits on the variability of G using binary-pulsar data havebeen obtained by Damour, Gibbons and Taylor [31]. A detailed analysis of Friedmann-Robertson-Walker (FRW)Universes in a wide range of scalar-tensor theories of gravity has been performed by Barrow and Parsons [26].

It is the purpose of the present paper to consider the evolution of a causal bulk viscous fluid filled flat FRW typeUniverse, by assuming the standard equations of state for the bulk viscosity coefficient, temperature and relaxationtime and in the presence of variable gravitational and cosmological constants. In order to obtain some very generalproperties of this cosmological model with variable constants we shall adopt a method based on the studies of thesymmetries of the field equations. As a first step we shall study the field equations from dimensional point of view.The dimensional method provides general relations between physical quantities and allow to make some definiteassumptions on the behavior of thermodynamical quantities. In particular we find that, under the assumption ofthe conservation of the total energy of the Universe, the bulk viscous pressure of the cosmological fluid must beproportional to the energy density of the matter component. With the use of this assumption the gravitational fieldequations can be integrated exactly, leading to a general solution in which all thermodynamical quantities have apower-law time dependence.

As we have been able to find a solution through dimensional analysis, it is possible that there are other symmetriesof the model, since dimensional analysis is a reminiscent of scaling symmetries, which obviously are not the mostgeneral form of symmetries. Hence we shall study the model through the method of Lie group symmetries, showingthat under the assumed hypotheses of the proportionality of bulk viscous pressure to the energy density, there are noother solutions of the field equations.

The present paper is organized as follows. The field equations are written down in Section II. The dimensionalanalysis of the model is performed in Section III. The general solution of the gravitational field equations for the bulkviscous pressure proportional to the energy density is obtained in Section IV. The Lie group symmetry study of themodel is considered in Section V. In Section VI we discuss and conclude our results.

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II. FIELD EQUATIONS, THERMODYNAMICS AND CONSEQUENCES

In the presence of a time variable gravitational and cosmological constants G and Λ the Einstein gravitational fieldequations are:

Rik −1

2gikR =

8πG(t)

c4Tik + Λ(t)gik. (1)

Applying the covariance divergence to the second member of equation (1) one obtain the generalized “conservationequation”

T ji;j =

(

−G,j

G

)

T ji −

c4Λ,i

8πG. (2)

We assume that the geometry of the space-time is of FRW type, with a line element

ds2 = c2dt2 − a2(t)(

dx2 + dy2 + dz2)

. (3)

The Hubble parameter associated with the line element (3) is defined as H = aa .

The energy-momentum tensor of the bulk viscous cosmological fluid filling the very early Universe is given by

T ki =

(

ρc2 + p + Π)

uiuk − (p + Π) δk

i , (4)

where ρ is the mass density, p the thermodynamic pressure, Π the bulk viscous pressure and ui the four velocitysatisfying the condition uiu

i = 1. The particle and entropy fluxes are defined according to N i = nui and Si =

eN i −(

τΠ2

2ξT

)

ui, with n is the number density, e the specific entropy, T ≥ 0 the temperature, ξ the bulk viscosity

coefficient and τ ≥ 0 the relaxation coefficient for transient bulk viscous effect (i.e. the relaxation time).The evolution of the cosmological fluid is subject to the dynamical laws of particle number conservation N i

;i = 0

and Gibb’s equation Tde = d(

ρc2

n

)

+pd(

1n

)

. In the following we shall also suppose that the energy-momentum tensor

of the cosmological fluid is conserved, that is T ki;k = 0.

For the evolution of the bulk viscous pressure we adopt the causal evolution equation [43], obtained in the simplest

way (linear in Π) to satisfy the H-theorem (i.e. for the entropy production to be non-negative, Si;i = Π2

ξT ≥ 0 [5]).

Therefore the gravitational field equations describing the cosmological evolution of a causal bulk viscous fluid inthe presence of variable gravitational and cosmological constants are

3H2 = 8πG(t)ρ + Λ(t)c2, (5)

2H + 3H2 = −8πG(t)

c2(p + Π) + Λ(t)c2, (6)

ρ + 3

(

ρ +p

c2+

Π

c2

)

H = 0, (7)

8πρG(t) + Λ(t)c2 = 0, (8)

τΠ + Π = −3ξH −ǫ

2τΠ

(

3H +τ

τ−

ξ

ξ−

T

T

)

. (9)

In eq. (9), ǫ = 0 gives the truncated theory (the truncated theory implies a drastic condition on the temperature),while ǫ = 1 gives the full theory. The non-causal theory has τ = 0. Dot denotes differentiation with respect to time.

In order to close the system of equations (5-9) we have to give the equation of state for p and specify T , ξ and τ .As usual, we assume the following phenomenological laws [43]:

p = (γ − 1)ρc2, ξ = αρs, T = βρr, τ = ξρ−1 = αρs−1, (10)

where α ≥ 0, β ≥ 0 are dimensional constants and 1 ≤ γ ≤ 2, s ≥ 0 and r ≥ 0 are numerical constants. Eqs. (10) arestandard in cosmological models whereas the equation for τ is a simple procedure to ensure that the speed of viscouspulses does not exceed the speed of light. These are without sufficient thermodynamical motivation, but in absenceof better alternatives we shall follow the practice adopting them in the hope that they will at least provide indicationof the range of possibilities. The temperature law is the simplest law guaranteeing positive heat capacity.

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With the use of equations of state (10) the evolution equation of the bulk viscous pressure becomes

Π +1

αρ1−sΠ = −3ρH −

1

2Π

[

3H − (1 + r)ρ

ρ

]

. (11)

In the context of irreversible thermodynamics p, ρ, T and the particle number density n are equilibrium magnitudeswhich are related by equations of state of the form ρ = ρ(T, n) and p = p(T, n). From the requirement that the entropy

is a state function, we obtain the equation(

∂ρ∂n

)

T= p/c2+ρ

n − Tn

[

∂∂T

(

p/c2)]

n. For the equations of state (10) this

relation imposes the constraint r = γ−1γ , so that 0 ≤ r ≤ 1/2 for 1 ≤ γ ≤ 2, a range of values which is usually

considered in the physical literature [7].The growth of the total comoving entropy Σ over a proper time interval (t0, t) is given by [43]:

Σ(t) − Σ (t0) = −3

kB

∫ t

t0

ΠHa3

Tdt, (12)

where kB is the Boltzmann’s constant.The Israel-Stewart-Hiscock theory is derived under the assumption that the thermodynamical state of the fluid is

close to equilibrium, that is the non-equilibrium bulk viscous pressure should be small when compared to the localequilibrium pressure |Π| << p = (γ − 1)ρc2 [44]. If this condition is violated then one is effectively assuming thatthe linear theory holds also in the nonlinear regime far from equilibrium. For a fluid description of the matter, thecondition ought to be satisfied.

An important observational quantity is the deceleration parameter q = ddt

(

1H

)

− 1. The sign of the decelerationparameter indicates whether the model inflates or not. The positive sign of q corresponds to “standard” deceleratingmodels whereas the negative sign indicates inflation.

By using the field equations we can express the gravitational constant in the form

G(t) =3

4π

HH

ρ. (13)

With the use of this expression for G we obtain the cosmological constant in the form

Λ(t)c2 = 3H2

(

1 −2Hρ

Hρ

)

. (14)

From the field equations we obtain for the derivative of the Hubble function the alternative expression

H = −4πG(t)

c2

(

ρc2 + p + Π)

. (15)

III. DIMENSIONAL ANALYSIS OF FIELD EQUATIONS

In this section we show how writing the field equations in a dimensionless way a particular solution of the fieldequations, corresponding to a specific choice of the bulk viscous pressure, can be obtained.

The π−monomia is the main object in dimensional analysis. It may be defined as the product of quantities whichare invariant under change of fundamental units. π − monomia are dimensionless quantities, their dimensions areequal to unity. The dimensional analysis has structure of Lie group [45]. The π − monomia are invariant under theaction of the similarity group. On the other hand we must mention that the similarity group is only a special classof the mother group of all symmetries that can be obtained using the Lie method. For this reason when one usesdimensional analysis only one of the possible solutions to the problem is obtained.

The equations (5-9) and the equation of state (10) can be expressed in a dimensionless way by the followingπ − monomia:

π1 =Gpt2

c2, π2 =

G |Π| t2

c2, π3 = Gρt2, π4 =

|Π|

p, π5 =

ξ

|Π| t, π6 =

τ

t= τH , (16)

π7 =ξ

αρs, π8 =

ξ

τρ, π9 =

T

βρr, π10 =

ρc2

p, π11 = Λc2t2. (17)

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The following relation can be obtained from the π − monomia. From π1, π2, π3, π4 and π10 we can see that

ρ ∝ p ∝ |Π| , (18)

note that [|Π|] = [p] =[

ρc2]

, that is, they have the same dimensional equation and therefore from this point of view

they must have the same behavior in order of magnitude. Now, from π7 = ξαρs and π8 = ξ

τρ we obtain

π8 =αρs−1

t, (19)

implying

ρ ∼

(

t

α

)1

s−1

, s 6= 1. (20)

From π3 and π8 we find

Gρt2

c2=

αρs−1

t, (21)

or

α1

s−1 c2

G= t

1−2s1−s , s 6= 1. (22)

If s = 1/2 we obtain the relationship α2 = c2/G. If s 6= 1/2 the “constant” G must vary.Therefore the solutions that Dimensional Analysis suggests us are the following ones:

ρ ∝(

α−1t)

1s−1 , p ∝

(

α−1t)

1s−1 = (γ − 1) ρc2, |Π| ∝

(

α−1t)

1s−1 = χρc2 (23)

Λ ∝ c−2t−2, G ∝ α1

s−1 c2t2s−11−s , s 6= 1, (24)

with χ a numerical constant. From the equation of state of bulk viscosity coefficient, temperature and relaxation timewe find

ξ ∝ α(

α−1t)

ss−1 , T ∝ β

(

α−1t)

rs−1 , τ ∝ t, s 6= 1. (25)

IV. GENERAL SOLUTION OF THE GRAVITATIONAL FIELD EQUATIONS WITH BULK VISCOUS

PRESSURE PROPORTIONAL TO THE ENERGY DENSITY

As we have seen in the previous Section, dimensional analysis suggests us that |Π| ∝ ρc2. Hence generally thebulk viscous pressure can be represented in the form Π = −χρc2, with χ ≥ 0 and χ ≪ 1. With this hypothesis theconservation equation becomes

ρ + 3 (γ − χ) ρH = 0, (26)

leading to the following relationship between the energy density ρ and the scale factor a:

ρ =ρ0

a3(γ−χ), (27)

where ρ0 > 0 is an arbitrary constant of integration. With the use of the assumed functional form of Π the evolutionequation of the bulk viscous pressure becomes

ρ = −K0ρ2−s, (28)

where we denoted K0 = 1α

[

1−r2 + 2−χ

2χ(γ−χ)

]

. The solution of this equation is

ρ(t) =ρ0

t1

1−s

, (29)

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with ρ0 = [(1 − s)K0]1/(s−1). From Eqs. (27) and (29) we obtain the time dependence of the scale factor

a(t) = a0t1

3(1−s)(γ−χ) . (30)

with a0 =(

ρ0

ρ0

)1

3(γ−χ)

.

The Hubble parameter is

H(t) =H0

t, (31)

where H0 = 13(1−s)(γ−χ) > 0. The behavior of the gravitational and cosmological constant can be obtained from Eqs.

(13) and (14), respectively:

G(t) =G0

t1−2s1−s

, Λ (t) =Λ0

t2, (32)

where G0 = 34π

H20 (1−s)

ρ0and Λ0 = 3H2

0 (2s − 1) /c2. The behavior of the bulk viscosity coefficient, temperature and

relaxation time is given by

ξ =ξ0

ts

1−s

, T =T0

tr

1−s

, τ = τ0t, (33)

where we denoted ξ0 = αρs0, T0 = βρr

0 and τ0 = ξ0/ρ0. The expression of τ is in agreement with the one expectedfrom a theoretical point of view, as argued in [43], since for a viscous expansion to be non-thermalizing we should haveτ < t, for otherwise the basic interaction rate for viscous effects could be sufficiently rapid to restore the equilibriumas the fluid expands.

The deceleration parameter q behaves as:

q = 1/H0 − 1 = const. (34)

For values of the parameters s, γ and χ so that 0 < H0 < 1 the expansion of the Universe is non-inflationary, whilefor H0 > 1 we obtain an inflationary behavior.

The comoving entropy in this model is

Σ(t) − Σ (t0) = Σ0t1

1−s (1

γ−χ+r−1), (35)

where we denoted Σ0 =3(1−s)χρ0H0a3

0

kBT0( 1γ−χ

+r−1).

The ratio of the bulk viscous and thermodynamic pressure is given by

∣

∣

∣

∣

Π

p

∣

∣

∣

∣

=

∣

∣

∣

∣

χ

γ − 1

∣

∣

∣

∣

. (36)

Since χ is assumed to be a small number, the present model is thermodynamically consistent for the matter equationsof state of cosmological interest, describing high-density cosmological fluids, when bulk viscous dissipative effects areimportant.

We would like to point out that we have obtained the same results as in the previous section, at least in order ofmagnitude. By direct integration of the field equations we have been able of finding some of the numerical constantsdescribing the evolution of the Universe. This is a crucial issue at least for the cosmological constant. With thiscomplete solution it is observed that we have the special case s = 1/2 for which we obtain

G = const., Λ = 0, ρ ∝ t−2, a ∝ t2

3(γ−χ) , ..., Σ(t) − Σ (t0) ∝ t2(1

γ−χ+r−1). (37)

If for example we take γ = 4/3 (radiation predominance) then

Σ(t) − Σ (t0) ∝ t2(3

4−3κ− 3

4 ). (38)

We therefore see that the χ−parameter, i.e. the causal bulk effect, perturb weakly the FRW perfect fluid solution. Ifχ = 0 we recover the perfect fluid case, and then the comoving entropy is constant.

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If s 6= 1/2 then we obtain the following behavior for G and Λ: if 1/2 < s < 1 then G is a growing function on timewhile Λ > 0, and Λ ∝ t−2, if s < 1/2 and Λ < 0 or s > 1 and Λ < 0, then G is a decreasing function on time whileΛ ∝ t−2.

These results agree with the ones obtained in a recent work [46] where we studied through different routes, renor-malization group (RG), dimensional analysis and structural stability a full causal cosmological model. The mainresult that we obtained in [46] was that only and only for s = 1/2 we obtain that the fixed point of the RG equationcorresponds to a flat non-viscous FRW universe, i.e. a perfect fluid case. The same result is obtained under thestandard stability analysis performed in this work, showing in this way that only for the case s = 1/2 the causal bulkviscous model approximates in the long time limit the dynamics of a flat perfect fluid filled FRW Universe.

V. LIE SYMMETRIES OF THE MODEL

In this Section we are going to study the field equations through the symmetry method. As we have indicated inthe introduction dimensional analysis is just a manifestation of scaling symmetry, but this type of symmetry is not themost general form of symmetries. Therefore by studying the form of G(t) for which the equations admit symmetries,we hope to uncover new integrable models. But we shall see that under the assumptions made we only obtain theprevious solutions obtained in the above sections.

We start again with the assumption Π = −χρc2, with χ ≥ 0. The bulk viscosity evolution equation can then berewritten in the alternative form

1 − r

2

ρ

ρ+

1

αρ1−s = 3

(

1

χ−

1

2

)

H. (39)

Taking the derivative with respect to the time of this equation and with the use of Eq. (15) we obtain the followingsecond order differential equation describing the time variation of the density of the cosmological fluid:

ρ =ρ2

ρ− Dρ1−sρ + AG(t)ρ2, (40)

where D = 2(1−s)α(1−r) > 0 and A = 12π(γ−χ)(χ−2)

χ(1−r) < 0.

Equation (40) is of the general form.

ρ = f(t, ρ, ρ), (41)

where f(t, ρ, ρ) = ρ2

ρ − Dρ1−sρ + AG(t)ρ2.

We are going now to apply all the standard procedure of Lie group analysis to this equation (see [47] for detailsand notation)

A vector field X

X = ξ(t, ρ)∂t + η(t, ρ)∂ρ, (42)

is a symmetry of (40) if

−ξft − ηfρ + ηtt + (2ηtρ − ξtt) ρ + (ηρρ − 2ξtρ) ρ2 − ξρρρ3+

+ (ηρ − 2ξt − 3ρξρ) f −[

ηt + (ηρ − ξt) ρ − ρ2ξρ

]

fρ = 0. (43)

By expanding and separating (43) with respect to powers of ρ we obtain the overdetermined system:

ξρρ + ρ−1ξρ = 0, (44)

ηρ−2 − ηρρ−1 + ηρρ − 2ξtρ + 2Dξρρ

1−s = 0, (45)

2ηtρ − ξtt + Dξtρ1−s − 3AξρGρ2 − 2ηtρ

−1 + D (1 − s) ηρ−s = 0, (46)

−AξGρ2 − 2AηGρ + ηtt + (ηρ − 2ξt) AGρ2 + Dηtρ1−s = 0, (47)

Solving (44-47), we find that

ξ(ρ, t) = −m(1 − s)t + b, η(ρ, t) = mρ (48)

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with the constraint

G

G=

(2s − 1)m

m(1 − s)t − b, (49)

and m, b are numerical constant.Thus we have found all the possible forms of G for which eq. (40) admits symmetries. There are two cases with

respect to the values of the constant m, m = 0 which correspond to G = const. and m 6= 0 which correspond to

G = G1 [m(1 − s)t + b]2s−11−s , (50)

with G1 a constant of integration. If s = 12 then G is a constant. For this form of G eq. (40) admits a single symmetry

X = (m(1 − s)t − b) ∂t − (mρ) ∂ρ. (51)

The knowledge of one symmetry X might suggest the form of a particular solution as an invariant of the operatorX , i.e. the solution of

dt

ξ (t, ρ)=

dρ

η (t, ρ)(52)

This particular solution is known as an invariant solution (generalization of similarity solution). In this case

ρ = ρ0t− 1

1−s , (53)

where for simplicity we have taken b = 0, and ρ0 is a constant of integration.We can apply a pedestrian method to try to obtain the same results. In this way, taking into account dimensional

considerations, from the eq. (40) we obtain the following relationships between density, time and gravitationalconstant:

α−1(1 − s)ρ1−st ⋍ 1, AGρt2 ⋍ 1. (54)

This last relationship is also known as the relation for inertia obtained by Sciama. From these relationship weobtain

ρ ≈ A−1G−11 [m(1 − s)t + b]

1−2s1−s t−2

≈ A−1G−11 [m(1 − s)t]

− 11−s . (55)

We can see that this result verifies the relation

α−1(1 − s)ρ1−st ⋍ 1. (56)

Therefore the conditions Π = −χρc2 together with the equation (11) are very restrictive. Hence we have shownthat under these assumptions there are no another possible solutions for the field equations.

VI. CONCLUSIONS

In the present paper we have studied a causal bulk viscous cosmological model, with bulk viscosity coefficientproportional to the energy density of the cosmological fluid. This hypothesis is justified by the dimensional analysis ofthe model. We have also assumed that the cosmological and gravitational constants are functions of time. With thehelp these assumptions the general solution of the gravitational equations can be obtained in an exact form, leadingto a power-law time dependence of the physical parameters on the cosmological time. The unicity of the solution isalso proved by the investigation of the Lie group symmetries of the basic equation describing the time variation ofthe mass density of the Universe.

In the considered model the evolution of the Universe starts from a singular state, with the energy density, bulkviscosity coefficient and cosmological constant tending to infinity. At the initial moment t = 0 the relaxation time isτ = 0. Generally the behavior of the gravitational constant shows a strong dependence on the coefficient s enteringin the equation of state of the bulk viscosity coefficient. For s = 1/2, G is a constant during the cosmologicalevolution. From the singular initial state the Universe starts to expand, with the scale factor a an s-dependentfunction. Depending on the value of the constants in the equations of state, the cosmological evolution is non-inflationary (q > 0 for H0 < 1) or inflationary (q < 0 for H0 > 1). Due to the proportionality between bulk viscous

9

pressure and energy density, both inflationary and non-inflationary models are thermodynamically consistent, withthe ratio of Π and p also much smaller than 1 in the inflationary era. Generally, the cosmological constant is adecreasing function of time. The present model is not defined for s = 1, showing that in the important limit of smalldensities an other approach is necessary.

Bulk viscosity is expected to play an important role in the early evolution of the Universe, when also the dynamicsof the gravitational and cosmological constants could be different. Hence the present model, despite its simplicity,can lead to a better understanding of the dynamics of our Universe in its first moments of existence.

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